Sedimentary formations generally exhibit slowly varying lateral changes in their lithological interfaces and physical properties. In some state-of-the-art logging-while-drilling (LWD) inversion methods, such as those used to determine formation resistivity based on resistivity data, the measured data are inverted on a point-by-point or sliding window basis using one-dimensional (1D) resistivity models. Predicted data and sensitivities are evaluated using semi-analytical solutions for a given set of model parameters defining the 1D resistivity model (e.g., layer thickness, resistivity, anisotropy ratio, relative dip, relative azimuth). The model parameters are then optimized such that they minimize the error between measured and predicted data subject to any enforced regularization. These inverse problems are usually over-determined. The 1D resistivity models are then stitched together to form a two-dimensional (2D) resistivity image, sometimes known as “curtain plots” by those of ordinary skill in the art.
In some cases, LWD inversions based on 2D pixel-based resistivity models or three-dimensional (3D) voxel-based resistivity models have also been disclosed. Here, the inversions are based on 2D or 3D resistivity models discretized as area elements (pixels) or volume elements (voxels), and the predicted data and sensitivities are evaluated using finite-difference, finite-element, or volume integral equation methods. The model parameters in each pixel or voxel are then optimized such that they minimize the error between measured and predicted data subject to any enforced regularization. These inverse problems are usually under-determined. In the literature, these methods have only been applied to synthetic data associated with idealistic resistivity LWD systems in isotropic formations. The performance of these methods for anisotropic formations hasn't been disclosed. These inversions are highly dependent on the choice of regularization, such as the choice of a priori model and the choice of stabilizing functional. Resistivity models often contain resistivity gradients from which formation interfaces are difficult to discern with any degree of confidence.
In each case, the resulting resistivity models often contain geologically unrealistic artefacts arising from model simplicity or an inappropriate choice of regularization.